Finite Group Databasebeta

← #14532406

Subgroups — #14532406

Order 2176.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#63256C211yesmaximal, characteristic, complemented
2C2×#10587626C211yesmaximal, characteristic, complemented
3C2×#10587624C211yesmaximal, characteristic, complemented
4#11762718C214yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762711C211yesmaximal, characteristic, complemented
7#11762685C214yesmaximal, complemented
8C4×#63227C211yesmaximal, characteristic, complemented
9C2×C17×#124C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2214yes—
13C22×C22⋊D17C2212yes—
14C2×#63256—24no—
15C2×#63256—24no—
16C2×#63256—24no—
17C22×D68C2212yes—
18C22×D68C2211yescharacteristic
19C22×C4×D17C2212yes—
20C22×C4×D17C2211yescharacteristic
21C17×#124—24no—
22C2×C22⋊C4×C17C2214yes—
23C2×C42×C17C2211yescharacteristic
24C2×C22⋊Dic17C2212yes—
25#10587626—24no—
26#10587624—24no—
27C2×#63227C2212yes—
28C2×#63227C2214yes—
29C2×C4⋊Dic17C2211yescharacteristic
30C2×C4×Dic17C2211yescharacteristic
31#10587502C2214yes—
32#10587502C2212yes—
33C24×C17C2312yes—
34C23×D17C2312yes—
35C23×D17C2311yescharacteristic
36C2×Q8×C17—22no—
37C2×Q8×C17—22no—
38C2×Q8×C17D414yescomplemented
39C2×D4×C17—22no—
40C2×D4×C17—22no—
41C2×D4×C17—22no—
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—28no—
47C2×C22⋊D17—28no—
48C2×C22⋊D17—44no—
49C2×C22⋊D17—44no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53#63256—232no—
54C2×D68—24no—
55C2×D68—24no—
56C2×D68—22no—
57C2×D68—24no—
58C2×D68—22no—
59C2×D68D414yescomplemented
60C2×C4×D17—24no—
61C2×C4×D17—22no—
62C2×C4×D17—24no—
63C2×C4×D17—22no—
64C2×C4×D17—24no—
65C2×C4×D17D414yescomplemented
66C22⋊C4×C17—28no—
67C42×C17—22no—
68C22⋊Dic17—24no—
69#63227—24no—
70#63227—44no—
71C4⋊Dic17—22no—
72C4×Dic17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—24no—
77C23×C17—24no—
78C23×C17C2411yescharacteristic
79C22×D17—22no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—21no—
83C22×D17—24no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—24no—
87C22×D17C2×D414yes—
88C22×D17C2×D412yes—
89Q8×C17—28no—
90D4×C17—44no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C2×C4×C17C2×D414yes—
99C2×C4×C17C2×D412yes—
100C22⋊D17—416no—
101C22⋊D17—48no—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—24no—
108C2×Dic17C2×D412yes—
109C2×Dic17#4114yescomplemented
110D68—216no—
111D68—28no—
112C4×D17—216no—
113C4×D17—28no—
114#2431—171nomaximal, Sylow
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—44no—
119C22×C17—44no—
120C22×C17C22×D411yescharacteristic
121C22×C17C22×D411yescharacteristic
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4112yes—
124C22×C17H(5,𝔽2)12yes—
125C2×D17—24no—
126C2×D17—22no—
127C2×D17—22no—
128C2×D17—22no—
129C2×D17—28no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—44no—
134C2×D17—42no—
135C4×C17—24no—
136C4×C17—42no—
137C4×C17#12014yes—
138Dic17—22no—
139Dic17—24no—
140Dic17—42no—
141C22×#41—171no—
142C2×#127—171no—
143C2×#124D1711yescharacteristic, complemented
144C2×#121—171no—
145#392—174no—
146#391—171no—
147#390—171no—
148#386—174no—
149C4×C22⋊C4—171no—
150C2×C17—82no—
151C2×C17C2×#12011yescharacteristic
152C2×C17#53211yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#54612yes—
155C2×C17#55012yes—
156D17—42no—
157D17—44no—
158C2×#41—344no—
159C2×#41—344no—
160C2×#41—344no—
161C22×Q8C2×D1711yescharacteristic
162C22×D4—174no—
163C22×D4—172no—
164C22×D4—171no—
165C22×D4—172no—
166C22×D4C2×D1711yescharacteristic
167C23×C4—171no—
168C23×C4—172no—
169#127—344no—
170#124—24no—
171#121—344no—
172C2×C4⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—174no—
176C2×C22⋊C4C2×D1714yes—
177C2×C42—171no—
178C2×C42C2×D1711yescharacteristic
179#95—172no—
180#95—174no—
181C17#243111yescharacteristic, complemented, Sylow
182C24—171no—
183C24—172no—
184C24C22×D1712yes—
185#41—3432no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C22⋊D1714yescomplemented
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—174no—
193C2×D4—344no—
194C2×D4—344no—
195C2×D4—348no—
196C2×D4—344no—
197C2×D4—348no—
198C2×D4—342no—
199C2×D4—344no—
200C2×D4—342no—
201C2×D4—684no—
202C2×D4—684no—
203C22×C4—172no—
204C22×C4—171no—
205C22×C4—174no—
206C22×C4—172no—
207C22×C4—344no—
208C22×C4—344no—
209C22×C4—342no—
210C22×C4—344no—
211C22×C4—342no—
212C22×C4C22×D1712yes—
213C22×C4C22×D1712yes—
214C22×C4C22×D1711yescharacteristic
215C4⋊C4—342no—
216C22⋊C4—28no—
217C22⋊C4—344no—
218C22⋊C4—344no—
219C22⋊C4—684no—
220C42—22no—
221C42—342no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—172no—
228C23—174no—
229C23—342no—
230C23—341no—
231C23—342no—
232C23—344no—
233C23—344no—
234C23—342no—
235C23—342no—
236C23—341no—
237C23C23×D1711yescharacteristic
238Q8—28no—
239D4—44no—
240D4—3416no—
241D4—348no—
242D4—6816no—
243D4—688no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—172no—
250C2×C4—174no—
251C2×C4—3416no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—341no—
256C2×C4—348no—
257C2×C4—342no—
258C2×C4—344no—
259C2×C4D4×D1712yes—
260C2×C4D4×D1712yes—
261C2×C4C2×C22⋊D1714yes—
262C2×C4C2×C22⋊D1712yes—
263C22—44no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—344no—
269C22—342no—
270C22—342no—
271C22—342no—
272C22—348no—
273C22—682no—
274C22—684no—
275C22—682no—
276C22—682no—
277C22—682no—
278C22C2×D4×D1711yescharacteristic
279C22C2×D4×D1711yescharacteristic
280C22#41×D1712yes—
281C22#1058769612yes—
282C22C22×C22⋊D1711yescharacteristic
283C4—24no—
284C4—42no—
285C4—342no—
286C4—344no—
287C4—682no—
288C4#1058762314yes—
289C2—82no—
290C2—682no—
291C2—684no—
292C2C2×#1058762311yescharacteristic
293C2#1176335811yescharacteristic
294C2#1176337711yescharacteristic
295C2#1176345212yes—
296C2#1176347812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
hypercenterC2×#124D17
hypocenterC17#2431
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#63256C2semidirect
2. C2×#10587626C2semidirect
3. C2×#10587624C2semidirect
4. #11762718C2semidirect
5. #11762716C2semidirect
6. #11762711C2semidirect
7. #11762685C2semidirect
8. C4×#63227C2semidirect
9. C2×C17×#124C2semidirect
38. C2×Q8×C17D4semidirect
59. C2×D68D4semidirect
65. C2×C4×D17D4semidirect
109. C2×Dic17#41semidirect
143. C2×#124D17semidirect
181. C17#2431semidirect
188. C2×Q8C22⋊D17semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#632562
C2×#105876262
C2×#105876242
#117627182
#117627162
#117627112
#117626852
C4×#632272
C2×C17×#1242
#243117